The breast holds significant importance in women's lactation process. The breast has diverse tissues such as glandular, Cooper’s ligament, lactiferous ducts, adipose, muscles, and skin with various mechanical properties. Most soft tissues exhibit nonlinear, viscoelastic, and anisotropic mechanical behaviour. Stiffness is one of the commonly used mechanical properties for measuring mechanical behaviour in soft tissues. However, this is typically used in simplifying cases where analysis is limited to within the linear elastic region. Young’s modulus is the prevailing measure for quantifying stiffness in tissues. This systematic review examines commonly used techniques for mechanically characterising breast soft tissues to close the gap in the literature. Additionally, this systematic review summarises the mechanical behaviour of the breast tissues available in the literature and highlights the uncovered areas to understand the mechanical behaviour of breast tissues fully. These include a) more mechanical behaviour studies, allowing more accurate modelling for other applications, b) Viscoelastic analysis studies, and c) mechanical analysis on other breast tissues, such as the lactiferous duct, also known as the milk duct. The data obtained in the review can be used as inputs in computational models to analyse the behaviour of the breast tissues.